What is the projection of &lt; 6 , 5 , 3 &gt; onto &lt; 2 , - 1 , 8 &gt; ?

Question

What is the projection of       &amp;lt;    6    ,    5    ,    3    &amp;gt;   onto       &amp;lt;    2    ,    -    1    ,    8    &amp;gt;  ?

Saottah87e · Accepted Answer

Given             a      →        =    &amp;lt;    6    ,    5    ,    3    &amp;gt;   and             b      →        =    &amp;lt;    2    ,    -    1    ,    8    &amp;gt;  , we can find       p    r    o          j                        b          →                            a      →      , the vector projection of             a      →       onto             b      →       using the following formula:      p    r    o          j                        b          →                            a      →        =          (                                    a            →                    ⋅                      b            →                                                |                          b              →                        |                              )                      b        →                    |                  b          →                |            That is, the dot product of the two vectors divided by the magnitude of             b      →      , multiplied by             b      →       divided by its magnitude. The second quantity is a vector quantity, as we divide a vector by a scalar. Note that we divide             b      →       by its magnitude in order to obtain a unit vector (vector with magnitude of       1  ). You may have noticed that the first quantity is a scalar because we know that the dot product of two vectors always yields a scalar value.Therefore, the scalar projection of       a   onto       b   is       c    o    m          p                        b          →                            a      →        =                  a        ⋅        b                              |          b          |                    , also written             |      p      r      o              j                              b            →                                      a        →            |      Taking the dot product of the two vectors will be our first step.            a      →        ⋅          b      →        =    &amp;lt;    6    ,    5    ,    3    &amp;gt;    ⋅    &amp;lt;    2    ,    -    1    ,    8    &amp;gt;        ⇒          (      6      ⋅      2      )        +          (      5      ⋅      -      1      )        +          (      3      ⋅      8      )            ⇒    12    -    5    +    24    =    31  Then, by taking the square root of the sum of the squares of each component, we can determine the size of             b      →                  |              b        →            |        =                                        (                          b              x                        )                    2                +                              (                          b              y                        )                    2                +                              (                          b              z                        )                    2                                |              b        →            |        =                                        (            2            )                    2                +                              (            -            1            )                    2                +                              (            8            )                    2                          ⇒                  4        +        1        +        64              =                  69            And now we have everything we need to find the vector projection of             a      →       onto             b      →            p    r    o          j                        b          →                            a      →        =                  31                              69                      ⋅                  &amp;lt;        2        ,        -        1        ,        8        &amp;gt;                              69                          ⇒                  31        &amp;lt;        2        ,        -        1        ,        8        &amp;gt;            69            =          31      69        &amp;lt;    2    ,    -    1    ,    8    &amp;gt;  The coefficient can be distributed to each element of the vector and written as follows:      ⇒    &amp;lt;          62      69        ,    -          31      69        ,          248      69        &amp;gt;  The scalar projection of             a      →       onto             b      →       is just the first half of the formula, where       c    o    m          p                        b          →                            a      →        =                  a        ⋅        b                              |          b          |                    . Therefore, the scalar projection is             31                        69                    , which does not simplify any further, besides to rationalize the denominator if desired, giving                     31                              69                              69

What is the projection of <6,5,3> onto <2,-1,8>?

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